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A standard six-sided die is rolled $7$ times.  You are told that among the rolls, there was one $1,$ one $2$, one $3$, one $4$, one $5$, and two $6$s.  How many possible sequences of rolls could there have been?  (For example, 2, 3, 4, 6, 6, 1, 5 is one possible sequence.)

 Aug 9, 2024
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First, we need to calculate the total possible number of rolls. Since the dice is rolled 7 times, we have \(7!\) total rolls. 

 

However, since there are duplicates with two 6s being rolled, we must divide by 2! to avoid overcounting. 

 

Thus, we just have

\(7! / 2! = 2520\)

 

So the answer is 2520. 

 

Thanks! :)

 Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024

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